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Theorem undifss 3339
Description: Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
undifss  |-  ( A 
C_  B  <->  ( A  u.  ( B  \  A
) )  C_  B
)

Proof of Theorem undifss
StepHypRef Expression
1 difss 3108 . . . 4  |-  ( B 
\  A )  C_  B
21jctr 308 . . 3  |-  ( A 
C_  B  ->  ( A  C_  B  /\  ( B  \  A )  C_  B ) )
3 unss 3156 . . 3  |-  ( ( A  C_  B  /\  ( B  \  A ) 
C_  B )  <->  ( A  u.  ( B  \  A
) )  C_  B
)
42, 3sylib 120 . 2  |-  ( A 
C_  B  ->  ( A  u.  ( B  \  A ) )  C_  B )
5 ssun1 3145 . . 3  |-  A  C_  ( A  u.  ( B  \  A ) )
6 sstr 3016 . . 3  |-  ( ( A  C_  ( A  u.  ( B  \  A
) )  /\  ( A  u.  ( B  \  A ) )  C_  B )  ->  A  C_  B )
75, 6mpan 415 . 2  |-  ( ( A  u.  ( B 
\  A ) ) 
C_  B  ->  A  C_  B )
84, 7impbii 124 1  |-  ( A 
C_  B  <->  ( A  u.  ( B  \  A
) )  C_  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    \ cdif 2979    u. cun 2980    C_ wss 2982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995
This theorem is referenced by:  difsnss  3551  exmidundif  3991  undifdcss  6467
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